IMA Postdoc Seminars are given weekly throughout the fall and spring semesters. Postdocs present on a variety of mathematical topics that may be unrelated to the current annual program theme. IMA visitors and University of Minnesota faculty are also invited to present on subjects of interest.

Zach Hamaker of the University of Minnesota will be organizing the 2014-2015 seminar series.

Titles, abstracts, and speakers for each seminar will be posted as available.
All seminars are from 2:00pm -
3:00pm unless otherwise noted.

The Unconditional Case of the S-Inequality for Exponential Measure

Abstract

Let m be the exponential distribution, i.e., the density of m is equal to 1/2^n exp(-(|x_1|+...+|x_n|)). Let K be an unconditional convex set and let P be a strip of the form [-p,p] times R^{n-1} with m(K)=m(P). We prove that m(tK) geq m(tP) for t>1.

This is a joint work with Tomasz Tkocz (University of Warwick).

Three Coloring the Discrete Torus
or
3-States Anti-Ferromagnetic Potts Model in Zero Temperature

Abstract

We prove that a uniformly chosen proper three coloring of Z_{2n}^d has a very rigid structure when the dimension d is sufficiently high.
In particular the coloring asymptotically almost surely takes one color on almost all of either the even or the odd sub-lattice.
This implies for example that one color appears on nearly half of the lattice sites.

This model is the zero temperature case of the 3-states anti-ferromagnetic Potts model, which has been studied extensively in statistical mechanics.
The result improves an independent bound due to Galvin, Kahn, Randall and Sorkin. The proof, however, is quite different:
using combinatorial methods which follow an algebraic-topological intuition, results of Peled about homomorphism height functions are extended to a new setting.

Joint work with Ron Peled.

Perfect Matchings in Dense Uniform Hypergraphs

Abstract

In graph/hypergraph theory, perfect matchings are fundamental objects of study. Unlike the graph case, perfect matchings in hypergraphs have not been well understood yet. It is quite natural and desirable to extend the classical theory on perfect matchings from graphs to hypergraphs, as many important problems can be phrased in this framework, such as Ryser's conjecture on transversals in Latin squares and the Existence Conjecture for block designs. I will focus on Dirac-type conditions (minimum degree conditions) in uniform hypergraphs and discuss some recent progresses. In particular, we determine the minimum codegree threshold for the existence of a near perfect matching in hypergraphs, which confirms a conjecture of Rödl, Ruciński and Szemerédi, and we show that there is a polynomial-time algorithm that can determine whether a k-uniform hypergraph with minimum codegree n/k has a perfect matching, which solves a problem of Karpiński, Ruciński and Szymańska completely.

The Brunn-Minkowski Inequality - Its Refinements and Extensions

Abstract

The Brunn-Minkowski inequality, which states that for every non-empty
compact sets $A,B$ in $\R^n$ and every $\lambda \in [0,1]$ one has
$$ |(1-\lambda)A + \lambda B|^{1/n} \geq (1-\lambda)|A|^{1/n} + \lambda
|B|^{1/n}, $$
where $|.|$ denotes the volume (Lebesgue measure), is a fundamental
inequality in mathematics.
The aim of this talk is to present this inequality together with its
consequences, refinements and extensions.

Frozen Random Walk

Abstract

We explore a variant of the symmetric random walk on $\mathbb{Z}$ where particles are frozen if they are on the extreme quarter on any of the two sides. The motivation of this process arises from a theoretical computer science result related to the algorithm giving a 2 coloring of an $n$ set with all discrepancies less than $6 \sqrt{n}$. We are interested in the maximum of the process and the mass distribution. Related to this is a deterministic process where we start with mass $1$ at the origin, and at each step we freeze $1/2$ of the extremal mass and we split the remaining mass at each discrete point equally among its neighbors. Under the assumption that the scaled distribution converges, we determine the distribution, and the maximum $q\sqrt{t}$ where $q$ satisfies $\frac{1}{2}q^2=\frac{qe^{-q^2/2}}{\sqrt{2\pi}(\Phi(q)-\Phi(-q))}$. We present various simulation results supporting the claims of the existence of a scaling limit as well as the connection between the random and the deterministic process. We emphasize that in the deterministic case, the limit shape of the mass distribution is not a truncated Gaussian at the expected endpoints.
Joint work with Shirshendu Ganguly, Yuval Peres and Joel Spencer.

Bijections for Reduced Decompositions

Abstract

We discuss enumerative theory for reduced decompositions of permutations and signed permutations. Particular focus will be paid to the Little map and insertion algorithms. These bijections allow for a refined study of combinatorial statistics for reduced decompositions.

Zykov's Symmetrization for Multiple Graphs with an Application to Erdos' Conjecture on Pentagonal Edges

Abstract

Erd\H{o}s, Faudree, and Rousseau (1992) showed that a graph on $n$ vertices and at least $\lfloor n^2/4\rfloor+1$ edges has at least \lfloor n/2\rfloor+1$ edges on triangles. This result is sharp, just add an extra edge to the complete bipartite graph. In this talk, we give an asymptotic formula for the minimum number of edges contained on triangles in a graph having $n$ vertices and $e$ edges. The main tool of the proof is a generalization of Zykov's symmetrization that can be applied for several graphs simultaneously. We apply our weighted symmetrization method to tackle Erd\H{o}s' conjecture concerning the minimum number of edges on 5-cycles. Many problems remain open.
This is a joint work with Zolt\'{a}n F\"{u}redi.

Biclique Decomposition of Random Graphs

Abstract

The biclique partition number bp(G) is the minimum number of complete bipartite graphs needed to partition the edges of a graph G. It is not hard to see that bp(G) <= n-\alpha(G), where \alpha(G) is the independence number. Erdős conjectured that for the random graph G=G(n, 0.5), bp(G)=n-\alpha(G) with high probability. In this talk I will discuss some recent progress and and remaining challenges in this area, and show that actually there exists an absolute constant c>0 such that for G=G(n, 0.5), bp(G) <= n-(1+c)\alpha(G) with high probability. Joint work with Noga Alon and Tom Bohman.

Localization in Infinite Dimensions

Abstract

In the context of Convex Geometry or Probability, the term localization
refers to a technique which reduces an n-dimensional integral inequality on R^n
to a related one dimensional inequality on R. Localization is particularly of
interest in the context of high-dimensional phenomena, as it obtains results
solely in terms of a parameter of the measure's "concavity". We will discuss
a generalizations of this technique to infinite dimensional spaces, as well as
applications.

Enumeration of Lozenge Tilings of a Hexagon with Holes on Boundary

Abstract

MacMahon's classical theorem on the number of plane partitions that fit in a given box is equivalent to fact that the number of lozenge tilings of a semi-regular hexagon of side-lengths $a,b,c,a,b,c$ (in cyclic order) on the triangular lattice is equal to
\[\frac{H(a)H(b)H(c)H(a+b+c)}{H(a+b)H(b+c)H(c+a)},\]
where the hyperfactorial function $H(n)$ is defined by
\[H(n):=0!1!\dots(n-1)!.\]
We generalize MacMahon's theorem by enumerating the lozenge tilings of a hexagon with holes on its boundary. In addition, we investigate a $q$-enumeration of plane partitions that fit in a special box consisting of several connected rooms.

A Bayesian Optimization Algorithm for Functionals: Application to Materials Modeling

Abstract

In computational materials science, low-dimensional coarse-grained (CG) simulations are often used as surrogates for their more expensive fully-atomistic (i.e. high-dimensional) counterparts. Mathematically, CG simulations can be viewed as mappings from a function f(x), which describes interparticle forces, to a real number that describes some material property. Typically, the f(x) that yields best agreement between the CG and atomistic models is at best known to within some probability. In this talk, I present an iterative Bayesian-type algorithm that can be used to rapidly improve our knowledge of optimal f(x) under certain conditions on the functional map.

Generating Random Graphs in Biased Maker-Breaker Games

Abstract

In this talk we present a general approach connecting biased Maker-Breaker games and problems about local resilience in random graphs. Using this
method, we investigate the threshold bias b for which Maker can win certain (1 : b) games. We utilize this approach to prove new results and also to derive some known results about biased games.
Joint work with: Michael Krivelevich and Asaf Ferber

Probabilistic Symmetrization Inequalities from a Combinatorial Point of View

Abstract

We consider certain symmetrization inequalities relating the probabilities that the sum and difference of two i.i.d. random variables lie in certain measurable sets. There are close connections between optimal constants in such estimates and some problems in combinatorial geometry and extremal graph theory, such as the kissing number problem and Tur\'{a}n theorem for triangle-free graph. As regards applications, such estimates are used to develop H\"{o}lder-type and reverse H\"{o}lder-type inequalities for certain class of random variables.

Catalan Matroid Decompositions of Certain Positroids

Abstract

Given a permutation $w$ in $S_n$, the matroid of a generic $n \times n$ matrix whose non-zero entries in row $i$ lie in columns $w(i)$ through $n+i$ is an example of a positroid. We enumerate the bases of such a positroid as a sum of certain products of Catalan numbers, each term indexed by the 3$-avoiding permutations above $w$ in Bruhat order. We also give a similar sum formula for their Tutte polynomials. These are both avatars of a structural result writing such a positroid as adisjoint union of a direct sum of Catalan matroids (up to isomorphism) and free matroids.

Winding of Planar Stationary Gaussian Processes

Abstract

Consider the path of a shift-invariant random map from the real numbers to the complex plane whose finite marginal distributions are Gaussian. Such processes are used by physicists to model polymers and random flux lines of magnetic fields. We investigate the winding number of such paths around the origin. We give exact forumlae for the mean and variance of this quantity, and prove a Central limit theorem. In doing so, we give rigorous proofs to predictions by physicists, such as Le Doussal, Etzioni and Horovitz.

Avoiding Repetitions on the Plane

Abstract

Inspired by classical Hadwiger-Nelson problem on chromatic number of R^2, we want to determine the number of colors required to avoid repetitions
on R^2. Since it turns out that in the above problem we need infinitely
many colors, we relax the problem. Namely, we show that 53 colors is enough to
avoid repetitions on paths of collinear points. On the other hand, we show what can be avoided using only 2 colors. Joint work with M. Dębski, J. Grytczuk, U. Pastwa, J. Sokół, M. Tuczyński, P. Wenus, K. Węsek.

The Frog Model on Trees

Abstract

Imagine that every vertex of a graph contains a sleeping frog. At time 0, the frog at some designated vertex wakes up and begins a simple random walk. When it lands on a vertex, the sleeping frog there wakes up and begins its own simple random walk, which in turn wakes up any sleeping frogs it lands on, and so on. This process is called the frog model.
I'll talk about a question posed by Serguei Popov in 2003: On an infinite d-ary tree, is the frog model recurrent or transient? That is, is each vertex visited infinitely or finitely often by frogs? Equivalently, do all frogs wake up eventually? The answer is that it depends on d: there's a phase transition between recurrence and transience as d grows. Furthermore, if the system starts with Poi(m) sleeping frogs on each vertex independently, there's a phase transition as m grows. This is joint work with Christopher Hoffman and Matthew Junge.

On the Perimeter of a Convex Set

Abstract

The perimeter of a convex set in R^n with respect to a given measure is the measure's density averaged against the surface measure of the set. It was proved by Ball in 1993 that the perimeter of a convex set in R^n with respect to the standard Gaussian measure is asymptotically bounded from above by n^{1/4}. Nazarov in 2003 showed the sharpness of this bound. We are going to discuss the question of maximizing the perimeter of a convex set in R^n with respect to any log-concave rotation invariant probability measure. The latter asymptotic maximum is expressed in terms of the measure's natural parameters: the expectation and the variance of the absolute value of the random vector distributed with respect to the measure. We are also going to discuss some related questions on the geometry and isoperimetric properties of log-concave measures.

The Brunn-Minkowski Inequality: Its Refinements and Extensions Part II

Abstract

My talk is related to the Brunn-Minkowski inequality, which states that for every convex sets $A,B$ in $\mathbb{R}^n$ and for every $\lambda \in [0,1]$, one has
$$ |(1-\lambda)A + \lambda B|^{1/n} \geq (1-\lambda) |A|^{1/n} + \lambda |B|^{1/n}, $$
where $A+B = \{a+b ; a \in A, b \in B \}$ denotes the Minkowski sum of $A$ and $B$ and where $|\cdot|$ denotes the volume (Lebesgue measure).
I will introduce a generalization of the Minkowski sum and prove a Brunn-Minkowski-type inequality for general measures, with respect to this new sum.
This is related to the log-Brunn-Minkowski inequality of Lutwak-Yang-Zhang

The MDS Conjecture

Abstract

The motivating question behind the MDS conjecture is: what is the maximum size g(k,q) of a family of vectors in (F_q)^k (the k-dimensional vector space over the finite field F_q) so that any k vectors in the family form a basis of (F_q)^k? This question interests the coding theory, algebraic geometry, and finite geometry communities, and its importance is highlighted by a 00 prize offered for its solution by the Information Theory and Applications (ITA) center at UCSD (http://media.itsoc.org/isit2006/vardy/handout.pdf). We give an exposition using the language of matrices of Simeon Ball's recent solution to this question when q is prime. If time permits, we will discuss joint work in progress on this question with Simeon Ball and Jan De-Buele.